[R] predict.lm - standard error of predicted means?

From: <kehler_at_mathstat.dal.ca>
Date: Thu 21 Jul 2005 - 01:28:50 EST


Simple question.

For a simple linear regression, I obtained the "standard error of predicted means", for both a confidence and prediction interval:

x<-1:15
y<-x + rnorm(n=15)
model<-lm(y~x)
predict.lm(model,newdata=data.frame(x=c(10,20)),se.fit=T,interval="confidence")$se.fit

        1 2
0.2708064 0.7254615

predict.lm(model,newdata=data.frame(x=c(10,20)),se.fit=T,interval="prediction")$se.fit

        1 2
0.2708064 0.7254615

I was surprised to find that the standard errors returned were in fact the standard errors of the sampling distribution of Y_hat:

sqrt(MSE(1/n + (x-x_bar)^2/SS_x)),

not the standard errors of Y_new (predicted value):

sqrt(MSE(1 + 1/n + (x-x_bar)^2/SS_x)).

Is there a reason this quantity is called the "standard error of predicted means" if it doesn't relate to the prediction distribution?

Turning to Neter et al.'s Applied Linear Statistical Models, I note that if we have multiple observations, then the standard error of the mean of the predicted value:

sqrt(MSE(1/m + 1/n + (x-x_bar)^2/SS_x)),

reverts to the standard error of the sampling distribution of Y-hat, as m, the number of samples, gets large. Still, this doesn't explain the result for small sample sizes.

Using R.2.1 for Windows



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